Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1381 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,16)(14,15)(18,22)(19,26)(20,23)(24,28)(25,31)(27,30);(1,4,5,3)(2,6,7,8)(10,14)(11,16)(17,18,23,28,21,24,20,22)(19,25,30,32,27,31,26,29);(10,14)(13,15)(18,24)(22,28)(25,31)(29,32);(17,19,20,26,21,27,23,30)(18,25,22,29,24,31,28,32)Polycyclic presentation:
2^9|b^2=fg,c^2=ghi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72313626, #72313671, #72313939 |
| maximal normal subgroup of | #72313626, #72313671, #72313939 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6726746);